Resolving the complex role of enzymeconformational dynamics in catalytic functionUrmi Doshi, Lauren C. McGowan, Safieh Tork Ladani, and Donald Hamelberg1

Department of Chemistry and the Center for Biotechnology and Drug Design, Georgia State University, Atlanta, GA 30302-4098

Edited by* J. Andrew McCammon, University of California, San Diego, La Jolla, CA, and approved February 14, 2012 (received for review October 15, 2011)

Despite growing evidence suggesting the importance of enzymeconformational dynamics (ECD) in catalysis, a consensus on howprecisely ECD influences the chemical step and reaction rates isyet to be reached. Here, we characterize ECD in Cyclophilin A, awell-studied peptidyl-prolyl cis-trans isomerase, using normal andaccelerated, atomistic molecular dynamics simulations. Kineticsand free energy landscape of the isomerization reaction in solutionand enzyme are explored in unconstrained simulations by allowingsignificantly lower torsional barriers, but in no way compromisingthe atomistic description of the system or the explicit solvent.We reveal that the reaction dynamics is intricately coupled to en-zymatic motions that span multiple timescales and the enzymemodes are selected based on the energy barrier of the chemicalstep. We show that Kramers’ rate theory can be used to presenta clear rationale of how ECD affects the reaction dynamics and cat-alytic rates. The effects of ECD can be incorporated into the effec-tive diffusion coefficient, which we estimate to be about ten timesslower in enzyme than in solution. ECD thereby alters the preex-ponential factor, effectively impeding the rate enhancement. Fromour analyses, the trend observed for lower torsional barriers can beextrapolated to actual isomerization barriers, allowing successfulprediction of the speedup in rates in the presence of CypA, whichis in notable agreement with experimental estimates. Our resultsfurther reaffirm transition state stabilization as the main effect inenhancing chemical rates and provide a unified view of ECD’s rolein catalysis from an atomistic perspective.

cis-trans isomerization ∣ Cyclophilin A ∣ enzyme catalysis ∣enzyme dynamics ∣ Kramers’ rate theory

Enzymes accelerate reaction rates by several orders of magni-tude, allowing them to occur at timescales relevant for cellularfunctions (1). One of the long-standing issues in biochemistry ishow enzymes achieve this remarkable speedup. It is commonlyaccepted that the most dominating effect arises from significantreduction in the free energy barrier compared to the correspond-ing noncatalyzed reaction in solution. It is also well establishedthat this predominant effect is mainly electrostatic in nature(2, 3), which is more favorable for the transition state than thereactant or the product (1). However, to what degree and howother factors such as desolvation, steric strain, and enzyme dy-namics contribute to catalysis remains disputable. Of particularinterest is the role of enzyme dynamics in catalysis that has stirredconsiderable debate (4–11) partly because it has not been clearlydefined, leading to a semantic issue. Also, the link betweenenzyme dynamics and catalysis is difficult to address both experi-mentally and theoretically. Currently, the implications of enzymedynamics are from ensemble- and time-averaged experiments, asthe temporal behavior of every atom cannot be observed directly.Although standard molecular dynamics (MD) simulations canprovide an atomistic picture of enzyme dynamics, they are stillnot amenable to study catalytic reactions that usually occur inmilliseconds. Computational approaches that have investigatedthe effects of millisecond-timescale enzyme dynamics on the che-mical reaction have been possible only with the use of coarse-grained models (4, 10). NMR relaxation dispersion experimentsthat can probe microsecond-millisecond timescale motions have

detected backbone and side-chain motions in and around theactive site that occur on the same millisecond-timescale as thechemical step (12). It has been further shown that such slowmotions are already present in the free enzyme (13). Further-more, loss of conformational fluctuations occurring in millise-conds in the active site of mutant enzyme has been observedwith concomitant reduction in activity (11). Single moleculestudies on enzymes have also revealed that catalytic rates canfluctuate over five orders of magnitude—from milliseconds tohundreds of seconds, similar to the range of timescale for confor-mational fluctuations (14). These observations are not surprisinggiven that protein dynamics comprise motions that span multipletimescales and occur in either a more localized or collective man-ner (15, 16). Nevertheless, protein dynamics has been suggestedto directly contribute to catalytic function and rate enhancement.The exact nature of this dynamical contribution cannot be under-stood, unless specific questions regarding whether dynamicalmotions of enzymes help in lowering the activation barrier(i.e., barrier effects) or aid the substrate to surmount the barrier(i.e., prefactor effects) are addressed.

The energy landscape of proteins is characterized by severalenergy minima that represent conformational substates separatedby barriers of varying heights (17, 18). Simultaneous motions ofmany degrees of freedom constitute protein dynamics and bringabout equilibrium interconversions (16, 19). We sought to under-stand the role of enzyme conformational dynamics (ECD) incatalytic functions by employing a combination of normal MD(nMD) and accelerated MD (aMD) (20) approach that providesatomistic detail with extended timescale. We chose to study Cy-clophilin A (CypA) (Fig. 1A), an extensively studied peptidyl-pro-lyl cis-trans isomerase, that catalyzes isomerization of the peptide(ω) bond preceding proline residues in proteins. Such system isideal to study using classical molecular mechanics because nobond breaking or formation is involved in the catalytic process.

Probing the influence of CypA dynamics on thechemical stepThe uncatalyzed isomerization reaction (RO) is an extremely slowprocess with an activation barrier of approximately 20 kcal∕moland occurs readily in hundreds of seconds in solution (21, 22).Cyclophilins are known to accelerate prolyl isomerization by105–106 times, reducing the timescale to around milliseconds(23, 24). It is not feasible to simulate even the catalyzed reaction(RC) with nMD, since it is currently limited to only hundredsof nanoseconds. Therefore, to probe the effects of ECD in cat-alysis, we used several lowered torsional energy barriers aroundthe -Ala-Pro- ω bond of a well-studied substrate analogue,Ace-Ala-Ala-Pro-Phe-Nme (Fig. 1A). We then took advantage

Author contributions: D.H. designed research; U.D., L.C.M., S.T.L., and D.H. performedresearch; U.D., L.C.M., S.T.L., and D.H. analyzed data; and U.D. and D.H. wrote the paper.

The authors declare no conflict of interest.

*This Direct Submission article had a prearranged editor.

Freely available online through the PNAS open access option.1To whom correspondence should be addressed. E-mail: dhamelberg@gsu.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1117060109/-/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.1117060109 PNAS ∣ April 10, 2012 ∣ vol. 109 ∣ no. 15 ∣ 5699–5704

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of the linearity of the behavior to extrapolate to the desired bar-riers. This approach is similar to carrying out experiments in anoptimum temperature range and extrapolating to a temperatureoutside that range. The lower barriers allowed us to track thedwell times in the trans well before going over the barrier withsufficiently good statistics. Using nMD we investigated the ki-netics of prolyl isomerization in the free and the enzyme-boundsubstrate with the same value of the AMBER force field para-meterV 2 (seeMethods and SI Text, section 1.4), which is the maindeterminant of torsional barrier. We found that the decay of thesurvival probability function, SðtÞ, of dwell times for the referenceRO in the free substrate unambiguously exhibited single exponen-tial behavior (Fig. 1B). Progressively slower kinetics resulted asV 2 was systematically increased from 0 to 11 kcal∕mol. Interest-ingly, for RC, the resulting kinetics exhibited multiexponentialdecays (Fig. 1C).

In view of the notion that protein dynamics is not independentof its environment (19), we observed that peptide isomerizationdynamics was influenced by the environment, whether it be thefluctuations in the solvent or the active site of the environment.From Fig. 1 B andD, it became evident that in terms of timescale,the dynamics of the aqueous environment was relatively uniformwith the solvent motions occurring on a single or a very narrowtimescale. In contrast, enzymatic motions were dispersed over anextensive timescale and were coupled to substrate dynamics. Con-sequently, the different enzyme dynamic modes became apparentin the isomerization kinetics, yielding multiexponential behavior(Fig. 1 C and E). As the torsional barrier became progressivelygreater (with the increase in V 2), the distributions of timescalesshowed the following trend: the relative amplitudes of the fasterphases diminished and those of the slower phases showed a gra-dual increase. These results suggested that, depending on thebarrier, and hence characteristic timescale, of the chemical step,the reaction dynamics got coupled to the slightly faster and slowerenzyme motions, resulting in multiexponential decays. Overall,the time constant for RO in solution was always larger than theaverage lifetime of the chemical step in the enzyme-boundsubstrate, RC (Table I). It was evident from our results that thechemical step was coupled to, and would be affected by, the en-zyme motions.

Accelerating CypA Dynamics and Its Effects on theChemical StepAnother question of interest is whether the contribution fromECD can significantly enhance the rate of the chemical step ascompared to that in solution. In answering this question, we sub-jected only CypA to increasing levels of aMD with the substratestill stimulated with nMD (SI Text, section 1.3). In order to con-firm that aMD indeed resulted in faster ECD in CypA, we char-acterized the fluctuations in free CypA(Co) from independentnMD and aMD (SI Text, section 1.2 and 1.3). Accelerated MDbrought about an increase in not only conformational plasticity;i.e., greater amplitudes of fluctuations as depicted from the shiftof backbone (Fig. 2 A and B) and side chain (Fig. S1) order para-meters (S2) to lower values, but also conformational heterogene-ity (Fig. S2) at the active site of Co. Similar to our results, aMDhas been shown to successfully increase the rate of conforma-tional sampling, thereby characterizing millisecond-timescaleprotein/peptide dynamical motions and achieving notable agree-ment with experimental data (25–29). Our simulations furtherconfirmed recent experimental observations (13, 14) that ECDin CypA takes place over a broad range of timescales even inthe substrate-free state (Fig. S1). Accelerating CypA dynamicsclearly affected the kinetics of prolyl isomerization in the boundsubstrate, resulting in faster decay of the survival probability(Fig. 2C). The decays fitted to multiexponential functions withonly three phases as opposed to five phases in the nMD of CypA(Fig. 2D). Since the enzyme modes sped up, the relative contri-bution of the faster phases increased as slower phases becamefaster (Fig. 2D). The net result was a gradual speed up in the aver-age lifetimes as the extent of acceleration of CypA motions wasincreased (Table I).

Using Kramers’ Rate Theory to Explain the Effects of CypADynamicsAlthough the usage of traditional rate theories to explain enzymekinetics has been a contentious matter (8, 30, 31), our resultscould be rationalized within the framework of Kramers’ theory(32, 33) in the high friction regime:k ¼ ωoωbDeff

2πkBTexp

�−ΔG#kBT

�, where

k is the rate of escape from the trans well with curvature ωoover the free energy barrier ΔG# with curvature ωb, kB is theBoltzmann constant and T is the temperature. Deff is the effec-

Fig. 1. Structure of CypA and the influence of its dynamics on the kinetics of prolyl isomerization in its substrate. (A) CypA (gray) with its substrate Ace-Ala-Ala-Pro-Phe-Nme in the binding pocket. The binding cavity (VDW surface) is defined by ten residues (nonpolar (white): Phe60, Met61, Ala101, Ala103, Phe113,Leu122 and polar (cyan): Arg55, Gln63, Asn102, and His126) that are within 4 Å of the substrate’s Ala-Pro motif. Decay of probability of survival in the transwellas a function of time obtained from nMD simulations of (B) RO and (C) RC with V2 set to 11.0 (cyan), 9.0 (red), 7.0 (blue), 5.0 (yellow), 4.0 (green), and 0.0(magenta) kcal/mol. Continuous lines are fits to single exponential in (B) andmultiexponential functions in (C). (Inset to B) Plots of survival probability functionson an extended timescale when V2 ¼ 7.0, 9.0 and 11.0 kcal∕mol. Parameters from exponential fits of isomerization kinetics, i.e. the amplitudes and the timeconstants, τ, are plotted for (D) RO and (E) RC with the same color code as in (B) and (C).

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tive diffusion coefficient on a one-dimensional free energy profileand assumed to be independent of the reaction coordinate. Deffincorporates the effects of the environment, as well as those in-herent in proteins, for example, frictional, dynamical effects andenergetic roughness. Frictional effects arise from solvent viscosityand internal friction that impede protein motions. Dynamicaleffects originate from enzyme with inhomogeneous diffusivity,arising from ECD occurring on a wide continuum of timescales,or an aqueous medium that offers a more homogeneous environ-

ment with essentially single (or very narrowly distributed) diffu-sion coefficient. The substrate undergoes desolvation whilemoving into the active site of the enzyme from aqueous solvent,as a result the energetic roughness may reduce, leading to a lesshindered substrate (34). The above Kramers’ relation can be rear-

ranged in the log form; i.e., ln�

kωoωb

�¼ ln

�Deff

2πkBT

�− 1kBT ΔG

#.

From the plots of ln�

kv2ω

v2o ω

v2b

�vs.ΔG#;v2 (where rate constants, cur-

vatures, and free energy barrier heights correspond to various va-lues of V 2) with a well-defined slope of 1∕kBT, the relativecontributions from the preexponential factor and the barrier ef-fects were estimated for RO and RC. (Fig. 3). Clearly, there wasspeedup in the isomerization rate in each case of the enzyme-bound substrate as compared to the corresponding RO (Fig. 3A).The speedup was the consequence of two opposing effects: In-crease in rate as a result of the reduction in free energy barrier,for example, 237 times from a barrier reduction of 3.26 kcal∕molin case of V 2 ¼ 9.0 kcal∕mol, which was offset by approximately2.6 times due to the modification in the curvatures of the freeenergy profiles and by approximately 13 times due to the differ-ences in the Deff in solution and in enzyme-bound substrate,bringing the net rate enhancement to only about seven times.It can be seen from Fig. 3A that the y-intercept, from whichthe effective diffusion coefficient can be estimated, is smallerfor RC than RO (Table I). Therefore, ECD does not enhancebut rather hinders the rate enhancement. When the dynamicsof CypA was accelerated, we clearly noticed an increase in theisomerization rates (Table I). Enzymatic CD does not directlymodify the properties of the free energy landscape, as that sce-nario would violate Boltzmann statistics. However, for each levelof acceleration subjected on CypA, the solvated CypA-substratecomplex should be considered a distinct system associated with itsHamiltonian and characteristic free energy profile. And indeed,the curvatures and barrier heights were modified (Table I) whenECD was accelerated. As the levels of acceleration were raised,both ΔG# and Deff showed an increase (Fig. 3B) relative to thecase in which ECD was not sped up (i.e., RC). Noticeably, evenwith the highest level of acceleration, Deff was not significantlyfaster (i.e., by only approximately 1.7 times) than the one in aqu-eous solution, given the errors associated with the calculation ofquantities from logarithmic scale and MD simulations. Furtheracceleration of CypA dynamics could reach a limiting case wherethe integrity of the active site might be lost as a result of theunfolding of the enzyme brought about by very fast dynamical

Fig. 2. Effects of accelerating CypA dynamics on prolyl isomerization in thesubstrate. Distribution of order parameters (S2) (SI Text, section 1.8) obtainedfrom (A) nMD and (B) aMD of free CypA using the highest level of accelera-tion. CypA structure is color-coded based on the S2 values of each backboneN-H bond vector (see color scale). The active site residues are shownwith stickrepresentation. Fluctuational motions with the largest amplitudes are indi-cated by the smallest S2 (red) while those with the smallest amplitudes aredepicted with the largest S2 (blue). (C) Decay of probability of survival in thetrans well as a function of time when V2 ¼ 7 kcal∕mol. (D) Parameters of ex-ponential fits in C; i.e., amplitudes and time constants, τ, are plotted with thesame color code as in (C). Shown are the isomerization kinetics in the freesubstrate (cyan, S) when subjected to nMD and the enzyme-bound substratewhen CypA was subjected to nMD (blue, ES) as well as aMD at the lowest(orange, A1), intermediate (dark red, A2), and the highest (dark green,A3) level of acceleration. Continuous black lines are mono- or multiexponen-tial fits.

Fig. 3. Comparison of prolyl isomerization kinetics in the free and the enzyme-bound substrate. Kramers’ plots are shown in the form of ln (k∕ωoωb) vs. ΔG#.(A) nMD data points for RO (open blue circles) and RC (filled blue circles) when V2 ¼ 0; 4, 5, 7, 9, and 11 kcal∕mol. (B) Same plot as in (A). For clarity, only datapoints from nMD corresponding to V2 ¼ 7.0 kcal∕mol are shown. Also plotted are the data points for RC from aMD when the lowest (violet), intermediate(magenta), and the highest (red) level of acceleration are applied on CypA. All continuous lines are linear fits with slope ¼ 1∕kBT . (C) Same plot as in (A) withlinear fits that are extrapolated to higher free energy barriers. Data points for RO (open orange circle) and RC (filled orange circle) are shown when thereoptimized V2 ¼ 28 kcal∕mol was used in aMD and assumed to follow the corresponding linear trends (i.e., red and green lines, respectively). Gray dashedlines above the red and below the green lines (with the same slope ¼ 1∕kBT ) represent illustrative kinetic trends for RC in CypA mutants with faster and slowerdynamics, respectively than the wild-type enzyme. Horizontal and vertical dashed lines with arrows depict reduction in free energy barriers and speedup inisomerization rates (along with sharper curvatures), respectively.

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motions. Our analysis therefore suggested that altering motionsassociated with ECD made the enzyme’s active site environmentmore aqueous-like and directly affected the preexponential fac-tor. At the same time, the favorable interactions between CypAand its substrate were possibly perturbed such that the free energyheight of isomerization is increased relative to RC but still re-mains lower than that for RO.

In an independent aMD study, setting V 2 to the reoptimizedvalue of 28 kcal∕mol (35), we calculated the free energy profileswith the expected actual barriers for RO and RC (SI Text, section1.3, and Fig. S4). We estimated the rate constants (Table 1) cor-responding to the actual RO and RC from their respective linearfits in Kramers’ plots (Fig. 3C). RC showed a speed up of approxi-mately 4.5 × 105 times over RO, which was strikingly similar toexperimental estimates (23). The above result therefore validatedKramers’ theory in the analysis of prolyl isomerization kineticsand its catalysis. We would like to note from Fig. 3C. that in thecase of CypA, as one goes to the higher barrier regime, the re-lative and dominant contribution from the reduction in barrierheights to the rate enhancement will continue to increase whilethe difference inDeff between RC and RC will remain at approxi-mately 13 times.

Recently, ambient temperature X-ray and relaxation NMRstudies on CypA have shown that impediment of motions thathelp in the interconversion of conformational substates in a mu-tant enzyme is accompanied with the reduction in catalytic rate(36). It was therefore concluded that protein dynamic motionscontribute directly to the catalytic power of the enzyme. Such re-sults can now be explained with Kramers’ theory that allows us tounderstand the nature of the dynamical contribution. The CypA-mutant with slower dynamics implies that the isomerization reac-tion would take place on a distinct free-energy profile with barrierheights and curvatures that are different from the wild-type CypAand with an effective diffusion coefficient that is perhaps slowerthan the wild-type enzyme (Fig. 3C). As we show above, it isequally important to investigate the dynamical effects as wellas the free-energy barrier effects of the mutant, which, in mostcases, is missing from experimental analyses. The recent studieslinking enzyme dynamics to catalysis have focused on mutantswith either slower or total absence of fluctuations in the activesite as compared to the wild-type enzyme (11, 36). It would beinteresting to investigate mutants with faster dynamics (Fig. 3C)and observe whether the catalytic rates are enhanced or not andhow they compare with wild-type- and nonenzymatic rates.

Characterizing ECD in the Free and Bound CypAWe compared the dynamic motions in the active site of CypA inthe absence and the presence of the substrate. Three separatenMD were carried out for the CypA-bound complex (Fig. 1A),when the -Ala-Pro- peptidyl-prolyl ω-bond of the substrate,Ace-Ala-Ala-Pro-Phe-Nme, was allowed to fluctuate in the trans

(Ctrans), transition (CTS) and cis (Ccis) states (SI Text, section 1.2).No significant differences were observed in either the distributionof backbone dihedral angles (Fig. S5) or amplitudes of backboneamide bonds in the Co versus bound-CypA or between Ctrans,CTS, and Ccis (Fig. S7). However, the distribution of side-chaindihedrals of the active site residues (Fig. S6) and the fluctuationsof selected side-chain bonds (Fig. S8) showed notable differencesin the absence and the presence of the substrate. Prominently, wefound that the side-chain rotamers that were preferred in thebound state are already sampled in Co and there is simply aredistribution of rotameric population in the bound-CypA ascompared to Co. Conforming to earlier studies (37, 38), the cat-alytically important Arg55, which have been shown to form sta-bilizing electrostatic interactions with the substrate in the transi-tion state, exhibited side-chain motions that were of smalleramplitudes and much more restricted in CTS than the end states(Figs. S6 and S8). The trajectories of Co, Ctrans, CTS, and Cciswere analyzed using principal component analysis (PCA) that de-composes the fluctuations in the atomic coordinates into modesranked according to their relative contribution to the overall pro-tein motion (SI Text, section 1.6). Projection of the trajectoriesonto the first three modes that accounted for 90% of the totalfluctuations resulted in two-dimensional representation of themultidimensional phase space (Fig. 4 A to C). The conforma-tional space sampled by the active site residues inCo was not onlymuch larger than that in Ctrans, CTS, and Ccis, but also showedconsiderable overlap with them, indicating that certain fluctua-tions observed in the enzyme in the presence of the substrateare already preserved in the free enzyme. CTS occupied a muchmore restricted conformational space as compared to Ctrans andCcis, in agreement with the reduction in conformational hetero-geneity observed in side chains of active site residues in CTS; i.e.,mainly a single side-chain rotamer is populated for all active siteresidues in CTS (Fig. S6). Also, as compared to Co, Ctrans, or Ccis,the side-chain fluctuations in CTS showed decreased amplitudes;i.e., higher S2 values (Fig. S8). As seen in Fig. 4, the smaller en-semble of CTS consisted of conformations that were a subset ofboth Ctrans and Ccis ensembles, each of which also had its ownunique set of conformations. Such connectivity between the con-formations of the ensembles is important for multiple numbers ofpossible catalytic pathways. The above results clearly affirmed thepicture of conformational selection (39) wherein the substrateprefers to bind a subset of enzyme conformations and upon cat-alysis there is a shift toward the population of enzyme conforma-tions bound to the product.

Estimating Binding Free Energies of the CypA-substrateComplexesWe estimated the binding free energies from each snapshot ofCtrans, CTS, and Ccis (Fig. 4D and SI Text, section 1.7). In supportof earlier studies, Fig. 4D revealed that CypA has maximum

Table 1. Free energy barriers, diffusion and time constants for the uncatalyzed and catalyzed prolyl isomerization*

V2 (kcal/mol)

Ro Rc

ΔG# (kcal/mol) hτi (ns) Deff(deg2 ∕s) ΔG#(kcal/mol) hτi (ns) Deff(deg2 ∕s)nMD 0.0 0.81 0.05 18.1 × 1014 - 0.01 1.37 × 1014

nMD 4.0 2.3 0.28 18.1 × 1014 - 0.07 1.37 × 1014

nMD 5.0 3.16 0.54 18.1 × 1014 - 0.16 1.37 × 1014

nMD 7.0 3.93 2.48 18.1 × 1014 0.99 0.50 1.37 × 1014

nMD 9.0 5.57 16.11 18.1 × 1014 2.31 1.35 1.37 × 1014

nMD 11.0 7.42 61.65 18.1 × 1014 3.49 6.24 1.37 × 1014

aMD 28.0 20.0a 1.22 × 1011b 10.74a 2.75 × 105b

aMD I 7.0 2.09 0.26 5.443 × 1014

aMD II 7.0 2.37 0.14 13.362 × 1014

aMD III 7.0 2.7 0.09 31.229 × 1014

*ΔG#0 s were calculated from potentials of mean force obtained from umbrella sampling except for (a) where they were obtained after reweighting freeenergy profiles resulting from aMD. (b) Time constants extrapolated from linear fits (red and green lines) in figure 3c. AMD levels I, II and III are the lowest,intermediate and the highest extents of acceleration (same as in Fig. 2D) subjected only on CypA. Deff ¼ 2πkBT:expðlnðk∕wowbÞΔG#¼0Þ.

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affinity for the substrate in the transition state and binds thecis isomer more favorably than the trans isomer (23, 38). Theability of CypA to bind transition states better than the groundstates resulted in transition state stabilization by approximately10-13 kcal∕mol greater than the cis and the trans isomers, respec-tively (Fig. 4D). The estimates of binding free energy were, how-ever, approximate and did not include contributions from thechanges in configurational entropy (SI Text, section 1.7). As seenin Fig. S4 and Table 1, we indeed found that the trans to cis free-energy barrier height for the catalyzed isomerization of -Ala-Pro-ω bond was reduced by approximately 9.3 kcal∕mol. Thus, thetransition state was stabilized considerably greater as comparedto that of the cis or the trans isomers and this alone accounted fora speed up of approximately 6 × 106 in isomerization rate if thesame preexponential factor was assumed for RO and RC. How-ever, we showedDeff for RC to be approximately 13 times smallerthan that of RO. Our findings therefore also reinforce the idea ofselective binding and transition state stabilization as the majorbarrier effect in enhancing the rate of catalysis.

ConclusionsAlthough a growing body of experimental data suggests thatenzyme motions play an important role in its catalytic function,the exact nature of this dynamical contribution has never beenexplained earlier. Here, we show that substrate dynamics involvedin the chemical step, is coupled to the dynamics of the surround-ing medium which can either be the solvent or the active site ofthe enzyme. If the environment relaxes much faster than the che-mical step such that there is a clear separation of timescales, thenthe motions are not coupled; e.g., second-timescale cis-trans iso-merization in aqueous solution. Provided the timescale of thechemical step falls within that of the fluctuations in the environ-ment, not only the motions on the same timescale but also thosethat are slightly faster and slower than the chemical step get

coupled to substrate dynamics and result in multiexponentialkinetic decays. Since our studies were carried out in the regimeof lower free-energy barriers in which the timescale of substratedynamics was shifted to nanoseconds, we could observe the cou-pling with nanosecond-motions of the enzyme. But for the actualcis-trans isomerization that involves larger barriers and takesplace in milliseconds in CypA, increasingly slower modes of theenzyme clustered in the millisecond-timescale will be selected forcoupling with substrate dynamics. We further show that one-di-mensional Kramers’ theory in the high friction regime, which hasproved sufficiently valid in analyzing and explaining the kineticsof various problems of biological interest—which includes pro-tein folding (40–43) and cis-trans isomerization in peptides (25,29), can be equally useful to interpret the kinetics of enzyme-catalyzed chemical step and understand the role of enzymedynamics. The enzymatic motions do not and cannot modify thefree energy landscape of the enzyme-substrate complex, butrather reduce the effective diffusion coefficient as compared tothe reference nonenzymatic reaction in solution. These effectsthat are incorporated in the preexponential factor of the chemicalreaction reduce the speedup possible just from barrier effectsand, as we show, modify the diffusion coefficient by more thanan order of magnitude. Therefore, as often implied, enzyme dy-namics do not accelerate the chemical step. Enzymatic motionsare important for reorganization of the active site so that the tran-sition state is better stabilized. However, enzyme dynamics doesnot have to enhance the catalytic rates to be important to thechemical step. Perturbation of conformational motions causeddue to mutations or acceleration affects the catalytic rates result-ing from modification of both—the free energy profile and theeffective diffusion coefficient. In any case, it is the alterationof free-energy barriers that remains the dominant effect in eitherimpeding or enhancing catalytic rates. Our results provide themissing link between assertions made from theoretical studiesand observations from experimental studies, thereby unifying ap-parently disparate views about the role of enzyme dynamics incatalysis.

MethodsWe carried out an extensive characterization of CypA dynamics (includingbackbone of all residues and side chains of active site residues) in the absenceand presence of its substrate using all-atom nMD (SI Text). In order to accesslong-timescale motions, we further subjected CypA to aMD. For details onaMD methodology, see SI Text. We then characterized the isomerization re-action in solution and in enzyme both kinetically and thermodynamically. Theparameter (V2) in the AMBER force field is the force constant in the dihedralenergy functional ∑dihedrals

Vn2 ½1þ cosðnω-γÞ� where n, ω, and γ are the peri-

odicity, dihedral angle, and phase angle, respectively. V2 (for which n ¼ 2 andγ ¼ 180°) predominantly controls the rotational barrier only around the pep-tidyl-prolyl bonds with almost isoenergetic cis and trans states. Changing V2modifies the total potential and hence the free-energy barriers of isomeriza-tion. The default or reoptimized (35) value of V2 would yield such high bar-riers that sufficient number of trans-cis transitions will not be observed in astandard MD trajectory of even several hundred nanoseconds. Therefore, tomake the simulation of kinetics of cis-trans isomerization feasible and obtainreasonable statistics, we shifted the isomerization timescale from secondsto nanoseconds by reducing V2 to lower values of 11.0, 9.0, 7.0, 5.0, 4.0,and 0 kcal∕mol. It should be noted that V2 ¼ 0 kcal/mol does not imply zerotorsional barrier. With each value of V2, we performed individual nMD simu-lations on the free solvated substrate to model the uncatalyzed reaction insolution as well as on the CypA-substrate complex to model the catalyzedisomerization in the active site of the enzyme. In a different set of simula-tions, the CypA-bound substrate was simulated with nMD with V2 set to7.0 kcal∕mol while the enzyme was subjected to three increasing levels ofacceleration (SI Text, section 1.3). For each case, free energy profiles projectedonto the ω dihedral were calculated from umbrella sampling (SI Text, section1.5). ΔG#, ωo, and ωb were estimated from such one-dimensional free energyprofiles (Fig. S3). The probability of survival in the trans well for time t orlonger was then calculated from the distribution of dwell times, pðτÞ, as fol-lows: SðtÞ ¼ ∫ þ∞τ¼t pðτÞdt. We analyzed our results using the DISCRETE (44, 45)software program that provides nonlinear least square solution of multiex-ponential decays without any a priori guesses for the number of exponentials

Fig. 4. Conformational selection and transition state stabilization by CypA.(A, B, C) Projection of the multidimensional Cartesian space onto the firstthree principal components. Shown are the configurations of the bindingsite residues (see SI Text, section 1.6) resulting from the snapshots of a300-ns long trajectory of free CypA (black) and 50-ns long trajectory eachof Ctrans (blue), CTS (green), Ccis; (red) complexes. (D) Probability distributionof binding free energies, ΔGbind, of Ctrans, CTS, and Ccis with average values of−20.8, −34.1, and −24.3 kcal/mol, respectively. The color code is the same as inA–C (Inset) Binding site of CypA (gray VDW) in CTS where the substrate makeshydrogen bonds (dashed green lines) with active site residues.

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or initial parameters. The survival probability function, SðtÞ, was fitted to asum of discrete multiexponential decays; i.e.,SðtÞ ¼ ∑ni¼1 Ai expð−t∕τiÞ. Theaverage rate constant k ¼ 1∕hτi where the average lifetime hτi ¼ ∑ni Aiτi .Ai and τi are amplitudes and time constants, respectively, of phase i inthe above function. Using the MM/PBSA (SI Text, section 1.7) method, wefurther calculated the binding free energies of Ctrans, CTS, and Ccis to estimatethe contribution of transition state stabilization to the speedup in the iso-merization rate.

ACKNOWLEDGMENTS. This work was supported in part by the National ScienceFoundation CAREER Grant MCB- 0953061, the Georgia Cancer Coalition(GCC) scholar award and Georgia State University. This work was also sup-ported by Georgia State’s IBM System p5 supercomputer, acquired througha partnership of the Southeastern Universities Research Association and IBMsupporting the SURAgrid initiative. We thank Drs W. David Wilson, StuartAllison, and Tongye Shen for helpful discussions during the preparation ofthis manuscript.

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